Suppose $X =\{−4, −3, −2, −1, 0, 1, 2, 3, 4 \}$ and suppose $E[X] = 0$. Give an upper bound on $P (X = 4)$ and an upper bound on $P (X < −2)$.
Solution given as:
Let $Y = X + 4$, $E[Y ] = E[X] + 4 = 4$
$P(X ≥ 4) = P(Y ≥ 8) ≤ E[Y]/8 = 4/8$
What happened "$P(X ≥ 4) = P(Y ≥ 8)$"? How can we calculate for $P(X<-2)$?